Reduction of Radical Quantities
263. Before entering on the consideration of the rules for the addition, subtraction, multiplication and division of radical quantities, it will be necessary to attend to the methods of reducing them from one form to another.
First, to reduce a rational quantity to the form of a radical;
Raise the quantity to a power of the same name as the given root, and then apply the corresponding radical sign or index.
Ex. 1. Reduce a to the form of the n-th root.
The n-th power of a is an. (Art. 207.)
Over this, place the radical sign, and it becomes n√ .
It is thus reduced to the form of a radical quantity, without any alteration of its value. For n√ =an/n = a.
2. Reduce 3a to the form of the 4th root.
Ans. 4√ .
3. Reduce 3.(a - x) to the form of the cube root.
Ans. 3√ . See Art. 208.
4. Reduce a2 to the form of the cube root.
The cube of a2 is a6. (Art. 216.)
And the cube root of a6 is 3√ = (a6)1/3.
In cases of this kind, where a power is to be reduced to the form of the n-th root, it must be raised to the n-th power, not of the given letter, but of the power of the letter.
Thus in the example, a6 is the cube, not of a, but of a2.
5. Reduce a2b4 to the form of the square root.
6. Reduce am to the form of the n-th root.
264. Secondly, to reduce quantities which have different indices, to others of the same value having a common index;
1. Reduce the indices to a common denominator.
2. Involve each quantity to the power expressed by the numerator of its reduced index.
3. Take the root denoted by the common denominator.
Ex. 1. Reduce a1/4 and b1/6 to a common index.
1st. The indices 1/4 and 1/6 reduced to a common denominator, are 3/12 and 2/12. (Art. 143.)
2d. The quantities a and b involved to the powers expressed by the two numerators, are a3 and b2.
3d. The root denoted by the common denominator is 1/12.
The answer, then, is (a3)1/12 and (b2)1/12.
The two quantities are thus reduced to a common index, without any alteration in their values.
For by Art. 250, a1/4 = a3/12, which by Art. 254, = (a3)1/12.
And universally an = am/mn = (am)1/mn.
2. Reduce a1/2 and bx2/3 to a common index.
The indices reduced to a common denominator are 3/6 and 4/6.
The quantities then, are a3/6 and (bx)4/6 or (a3)1/6 and (b4x4)1/6.
4. Reduce (a + b)2 and (x - y)2/3. Ans. [(a + b)6]1/3 and [(x - y)2]1/3.
5. Reduce a1/3 and b1/5.
6. Reduce x2/3 and 51/2.
265. When it is required to reduce a quantity to a given index;
Divide the index of the quantity by the given index, place the quotient over the quantity, and set the given index over the whole.
This is merely resolving the original index into two factors, according to Art. 254.
Ex. 1. Reduce a1/6 to the index 1/2.
By Art. 159, (1/6):(1/2) = (1/6).(2/1) = 2/6 = 1/3.
This is the index to be placed over a, which then becomes
a1/3; and the given index set over this, makes it (a1/3)1/2, the answer.
2. Reduce a2 and x3/2 to the common index 1/3.
2:(1/3) = 2.3 = 6, the first index
(3/2):(1/3) = (3/2).3 = 9/2, the second index
Therefore (a6)1/3 and (x9/2)1/3 are the quantities required.
3. Reduce 41/2 and 31/3, to the common index 1/6.
Answer, (43)1/6 and (32)1/6.
266. Thirdly, to remove a part of a root from under the radical sign;
If the quantity can be resolved into two factors, one of which is an exact power of the same name with the root; find the root of this power, and prefix it to the other factor, with the radical sign between them.
This rule is founded on the principle, that the root of the product of two factors is equal to the product of their roots. (Art. 255.)
It will generally be best to resolve the radical quantity into such factors, that one of them shall be the greatest power which will divide the quantity without a remainder. If there is no exact power which will divide the quantity, the reduction cannot be made.
Ex. 1. Remove a factor from √8.
The greatest square which will divide 8 is 4.
We may then resolve 8 into the factors 4 and 2. For 4.2 = 8.
The root of this product is equal to the product of the roots of its factors; that is, √8 = √4.√2.
But √4 = 2. Instead of √4 therefore, we may substitute its equal 2. We then have 2.√2 or 2√2.
This is commonly called reducing a radical quantity to its most simple terms. But the learner may not perhaps at once perceive, that 2√2 is a more simple expression than √8.
2. Reduce √x = a.√x = a√x.. Ans. √ .√
3. Reduce √18. Ans. √9.2 = √9.√2 = 3√2.
4. Reduce n√b, or ab1/n.. Ans. an√
5. Reduce (54a6b)1/3. Ans. 3a2(2b)1/3.
6. Reduce √.
7. Reduce 3√.
267. By a contrary process, the coefficient of a radical quantity may be introduced under the radical sign.
1. Thus, an√b = n√ .
Here the coefficient a is first raised to a power of the same name as the radical part, and is then introduced as a factor under the radical sign.
2. a(x - b)1/3 = [a3.(x - b)]1/3 = (a3x - a3b)1/3.
3. 2ab(2ab2)1/3 = (16a4b5)1/3.